In mathematics, a Baire measure is a measure on the

σ-algebra In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra") is part of the formalism for defining sets that can be measured. In calculus and analysis, for example, σ-algebras are used to define the concept of sets with a ...

Baire set In mathematics, more specifically in measure theory, the Baire sets form a σ-algebra of a topological space that avoids some of the pathological properties of Borel sets. There are several inequivalent definitions of Baire sets, but in the most ...

s of a

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

whose value on every compact Baire set is finite. In compact

metric spaces In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are a general setting for ...

the

Borel set In mathematics, a Borel set is any subset of a topological space that can be formed from its open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets ...

s and the

s are the same, so Baire measures are the same as

Borel measure In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. ...

s that are finite on

compact set In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., i ...

s. In general Baire sets and Borel sets need not be the same. In spaces with non-Baire Borel sets, Baire measures are used because they connect to the properties of

continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...

s more directly.

Variations

There are several inequivalent definitions of

s, so correspondingly there are several inequivalent concepts of Baire measure on a topological space. These all coincide on spaces that are locally compact σ-compact

Hausdorff spaces In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint sets, disjoint neighbourhood (mathematics), neighbourhoods. Of the many separation ...

Relation to Borel measure

In practice Baire measures can be replaced by regular Borel measures. The relation between Baire measures and regular Borel measures is as follows: *The restriction of a finite Borel measure to the Baire sets is a Baire measure. *A finite Baire measure on a compact space is always regular. *A finite Baire measure on a compact space is the restriction of a unique regular Borel measure. *On compact (or σ-compact) metric spaces, Borel sets are the same as Baire sets and Borel measures are the same as Baire measures.

Examples

Counting measure In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinit ...

on the

unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysi ...

is a measure on the Baire sets that is not regular (or σ-finite). *The (left or right)

Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This Measure (mathematics), measure was introduced by Alfr� ...

on a

locally compact group In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are lo ...

is a Baire measure invariant under the left (right) action of the group on itself. In particular, if the group is an

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...

, the left and right Haar measures coincide and we say the Haar measure is

translation invariant In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation (without rotation). Discrete translational symmetry is invariant under discrete translation. Analogously, an operato ...

. See also

Pontryagin duality In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), ...

References

* Leonard Gillman and

Meyer Jerison Meyer Jerison (November 28, 1922 – March 13, 1995) was an American mathematician known for his work in functional analysis and rings, and especially for collaborating with Leonard Gillman on one of the standard texts in the field: ''Rings of C ...

, ''Rings of Continuous Functions'', Springer Verlag #43, 1960 {{Measure theory Measures (measure theory)